User:Phlsph7/Arithmetic

Definition

Arithmetic is the study of numbers. It is a branch of mathematics and investigates operations on numbers, such as addition, subtraction, multiplication, and division.

Arithmetic is study of numbers and operations on numbers.

In its widest sense, it investigates the origin of the concept of number, methods to calculate numbers, the distinction between different types of numbers, the properties of numbers, and the fundamental axioms governing numerical operations.

All arithmetic operations and theorems have their foundations in the axioms describing addition and multiplication.

Arithmetic investigates the manipulation of numbers.

In an even wider sense, the term "arithmetic" also includes operations performed on entities that are not numbers, like "matrix arithmetic" and "arithmetic of quadratic forms".

Some definitions restrict arithmetic to the study of natural numbers. But in a wider sense, it investigates other types of numbers as well, including negative integers, rational numbers, irrational numbers, real numbers, and complex numbers.

Arithmetic is closely related to number theory, which restricts itself to the study of the properties of integers and is often referred to as "higher arithmetic". However, the distinction between the two terms is not consistent in the academic literature. Some mathematicians include the discussion numbers other than integers in the field of number theory and some use the terms arithmetic and number theory as synonyms.

Numbers

Types

Arithmetic deals with different classes of numbers. They can be grouped into natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

Natural numbers include only positive integers, like 1, 17, and 389. The symbol for the set of natural numbers is N. The class of whole numbers includes all non-negative integers. It is identical to the natural numbers together with 0. Some mathematicians do not draw the distinction between natural and whole numbers by including 0 in the class of natural numbers. Integers additionally include whole numbers that are negative, like -3, -24, and -962. The symbol for the set of integers is Z.

A number is rational if it can be represented using the quotient of two integers. For example, the rational number 1/2 is expressed by dividing the integer 1, called the numerator, by the integer two, called the denominator. Other examples are 3/4 and 281/3. The symbol for the set of rational numbers is Q. Decimal fractions like 0.3 and 2.512 are a special type of rational numbers since their denominator is a power of 10. For examples, 0.3 is equal to 3/10, and 2.512 is equal to 2512/1000.

Irrational numbers are numbers that cannot be expressed through the ratio of two integers. They include many square roots, like ${\sqrt {2}}$, and numbers like π, which is used to express the circumference of a circle. The class of rational numbers together with the class of irrational numbers makes up the class of real numbers. The symbol for the set of real numbers is REven wider classes of numbers include complex numbers and quaternions.

Numbers can also be distinguished into cardinal and ordinal numerals. Cardinal numerals are numbers that denote quantity, such as one, two, and three. They answer the question "how many?". Ordinal numerals indicate position or order in a series like first, second, and third. They answer the question "what position?".

Numbering system

A numbering system is a way to represent quantities. An important distinction is between positional and non-positional numbering systems. For positional numbering systems, the position of a digit in a number determines its value. Positional numbering systems use a radix that acts as a multiplier of the different positions. In the common decimal system, this radix is 10. In it, the first digit is multiplied by 10⁰, the second digit is multiplied by 10¹, and so on. Because of this, the number 532 differs from the numbers 325 and 253 even though they have use same digits. Non-positional systems resort to different means to represent numbers, like counting fingers or adding several numbers together, as in the Roman number system.

The dominant system today is the decimal system. It is based on the Hindu-Arabic system and relies on the digits from 0 to 9.

Computer use a binary system that only uses two digits: 0 and 1.

Arithmetic operations

Arithmetic operations are ways of manipulating numbers.

The two fundamental arithmetic operations are addition and multiplication. Other operations like subtraction, division, or raising to a power can be understood derivatives or inverse operations of them. For example, to subtract three from nine is the same as to add minus three to nine.

The two fundamental mathematical operations in arithmetic are addition and multiplication.

Addition is an arithmetic operation in which two numbers, called the addends, are combined to form a larger number, known as the sum.

Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference.

Multiplication is an arithmetic operation in which two numbers, called the multiplicand and the multiplier, are combined to produce a third number, known as the product.

For the class of whole numbers, multiplication is the same a repeated addition. For example, 2 * 4 is the same as 2 + 2 + 2 + 2.

Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by the second number, known as the divisor. The result of this operation is called the quotient.

Laws

According to the commutative law, the order of the addends does not matter. For example, 2 + 4 is the same as 4 + 2. According to the associative law, the grouping of the addends does not matter. For example, (2 + 3) + 4 is the same as 2 + (3 + 4).

According to the closure law, the sum of any two real numbers is itself a real number.

According to the law of additive identity, adding zero to any number does not change the value of the number. For example, 5 + 0 is still 5.

According to the inverse additive property, adding the opposite (or additive inverse) of a number to the number itself results in zero. For example, 3 + (-3) equals zero.

The laws of commutativity, associativity, and closure also apply to multiplication.

According to the law of multiplicative identity, multiplying any number by one does not change the value of the number. For example, 5 * 1 is still 5.

According to the inverse multiplicative property, multiplying a number by its reciprocal (or multiplicative inverse) results in one. For example, 5 * (1/5) equals one.

The law of distributivity governs the relation between addition and multiplication. According to it, the multiplication of a number by a sum is equal to the sum of the multiplication of the number by each addend. For example, 2 * (3 + 4) is the same as (2 * 3) + (2 * 4).

quote: "The axioms related to the operations of addition and multiplication indicate that real numbers form an algebraic field. Four additional axioms assert that within the set of real numbers there is an order. One states that for any two real numbers, one and only one of the following relations is true: either a < b, a > b or a = b. Another suggests that if a < b, and b < c, then a < c. The monotonic property of addition states that if a < b, then a + c < b + c. Finally, the monotonic property of multiplication states that if a < b and c > 0, then a*c < b*c."

The the inverse additive and multiplicative properties do not apply to the class of whole numbers.

For integers, the inverse additive is present for integers, but the inverse multiplicative property does not.

Integer arithmetic

Integer arithmetic is the branch of arithmetic that deals with the manipulation of positive and negative whole numbers. The set of integers is represented by the symbol $Z$ and can be expressed as {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Simple one-digit operations can be performed by following or memorizing a table which presents the results of all possible combinations, like an addition table or a multiplication table.

+	0	1	2	3	4	...
0	0	1	2	3	4	...
1	1	2	3	4	5	...
2	2	3	4	5	6	...
3	3	4	5	6	7	...
4	4	5	6	7	8	...
...	...	...	...	...	...	...

Example: The addition with carry of two-digit integers

For operations on numbers with more than one digit, different techniques can be employed to calculate the result by using several one-digit operations in a row. For example, in the method addition with carries, the two numbers are written one above the other. Starting from the rightmost digit, each pair of digits is added together. The rightmost digit of the sum is written below them. If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to the left. This process is repeated until all digits have been added. A similar techniaue is used for subtraction: it also starts with the rightmost digits and uses a "borrow" or a negative carry for the column on the left if the result of the one-digit subtraction would be negative.

A common technique for multiplication is called long multiplication. This method starts by writing the multiplicant above the multiplier. The calculations begins by multiplying the multiplicant only with the rightmost digit of the multiplier and writing the result below, starting in the rightmost column. The same is done for each digit of the multiplier and the result in each case is shifted one position to the left. As a final step, all the individual products are added to arrive at the total product of the two numbers. A technique used for division is called long division.

Integer arithmetic is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For example, 7 divided by 2 is not a whole number but 3.5. One way to ensure that result is an integer is to round the number to the next whole number. This method, however, may lead to inaccuracies as the original value is altered. Another method is perform the division only partially and retain the remainder. For example, 7 divided by 2 is 3 with a remainder of 1.

Rational arithmetic

Rational arithmetic is the branch of arithmetic that deals with the manipulation of fractions. A fraction is a number that can be expressed as a the ratio of two integers. The set of rational numbers is represented by the symbol $Q$ and includes numbers like 1/2, -7/9, and 243/127.

Foundations

The axioms of arithmetic are the rule that governs how arithmetic operations work.

Addition and multiplication can be defined by three axioms: communitativity, associativity, and closure.

Foundations of arithmetic aim to provide a rigorous mathematical formulation of its principle. They further seek to arrive at a simple epistemological justification of arithmetic knowledge and to clarify the ontological status of numbers.

Axiomatizations of arithmetic are sets of fundamental mathematical principles that govern all arithmetic operations. They aim to provide a small set of basic premises from which all arithmetic truths can be inferred through deductive reasoning.

A well-known axiomatization of arithmetic of natural numbers are the Peano axioms. Their basic principles were first formulates by Richard Dedekind and later refined by Giuseppe Peano. Their only primitive terms are zero, natural number, and successor. The five Peano axioms determine these terms are related to each other:

1. 0 is a natural number.
2. For every natural number, there is a successor, which is also a natural number.
3. The successors of two different natural numbers are never identical.
4. 0 is not the successor of a natural number.
5. A set contains every natural number if it contains 0 and the successer of each number in the set.

Axioms of arithmetic provide an epistemological foundations of all arithmetic truths. There are disagreements about how mathematicians know about the axioms themselves. One view states that they are self-evident truths that require no further explanation. This view is not generally accepted since, at least in some cases, even judgments of self-evidence are fallible.

Another difficulty in this regard is that human knowledge of specific formulas, like "2 + 2 = 4", seems to be more certain than their knowledge of general abstract axioms supposed to ground the knowledge of the specific formulas.

History

Arithmetic has its origin in the need to count entities.

One early step in the development of arithmetic was the insight that numbers express abstract ideas that can be conceptualized independent of the real items the enumerate. For example, four oxen and four baskets both instantiate the number four.

The first organized methods of managing numbers appeared 4000 BCE in ancient Sumeria.

More elaborate systems appeared later in the ancient period in Egypt, Babylonia, India, and China. They included an understanding of the difference between whole numbers and fractions as well as a familiarity with basic arithmetic operations on these numbers.

Ancient civilizations did not yet possess the concept of zero. This concept was still absent in the Roman numbering system.

Arithmetic has its foundation in idea of counting. Important early historical developments were the development of positional number systems and the invention of symbol for the number zero.

Arithmetic evolved as a formalization and extension of the practice of counting. Early steps included the recognition of numbers as abstract concepts that can be instantiated by very different groups of entities, for example, that a group of four trees and a group of four cows both instantiate the same quantity.

The Sumerians were the first ancient civilization to develop basic forms of arithmetic(?). More advanced arithematic practices evolved in the ancient civilizations of Egypt, Babylonia, India, and China. They used them in the fields of trade and commerce.

The ancient Greeks not only used arithmetic as a commercial tool but developed the a first general theory of numbers in the third century BCE.

The Hindu-Arabic system, which is still used today, was developed in India about 1500 years ago. In the Medieval period, the Arabs brought it to Europe. As a positional number system, it held various advantages of the non-positional Roman numeral system dominant at that time and eventually replaced it.

The first positional number system was developed in Babylonia and had 60 as its base.

The movement of logicism developed in the late 19th and early 20th centuries as an attempt to ground all arithmetic truths in the laws of logic alone, i.e., by deducing them from the axioms of logic. An influential proponent of this position was Gottlob Frege. Despite many fruitful developments in this regard, the program ultimately failed since several additional axioms were required in addition to the axioms of logic.

Historically, arithmetic is the first mathematical practice. Some of its basic concepts and operations, like whole number, addition, and multiplication, as well as the usage of a numeral system can be found in all civilizations.

The mathematical concept of zero was first invented in ancient India.

Arithmetic was already used before the time of writing. The earliest written evidence are the Cahoon papyri and the Rhind papyrus from the time of roughly 2000 BCE.

Ancient Egyptians had a hieroglyphic system to represent natural numbers, addition, and subtraction. They also had methods of dealing with multiplication and fractions.

The Babylonians used a hexadecimal system.

The Ancient Greeks understood arithmetic as the abstract analysis of the properties of numbers. The study of operations on numbers, like methods of calculation, was considered a separate discipline.

Greek mathematicians distinguished between numbers and magnitude.

Euclid's Elements from the 3rd century BCE contains influential treatments of arithmetic problems, such as discussions of prime numbers and the Euclidean algorithm to find the greatest common divisor of two integers. He further discussed properties of arthmetic operations, like commutativity and distributivity.

Diophantus developed the idea of exponentiation in the 3rd century CE.

In the second century CE, Chinese mathematicians

In the 2nd century CE, Chinese mathematicians performed calculations with fractions and negative numbers and later also included roots.

An influential development in 5th-century India was the development of the decimal numerical system and remains in use still today. Their also had advanced knowledge in other areas of mathematics, for example, in relation to proportions and percentages and later also negative numbers.

Many of their insights were adapted by the Arabs, who later spread them to Europe during the Middle Ages.

In the ancient period, both Greeks and Chinese used abacuses to perform calculations in a much more efficient way than by counting fingers. Abacuses became very common in medieval Europe.

In the 13th century, Leonardo da Pisa made various innovations, including a discussion of the addition of fractions using the least common multiple of the denominators and formulating calculations in positional systems other than base 10 as well as arithmetic and geometric progressions. He also introduced the problem of negative numbers to Europe.

In the 15th and the following centuries, new techniques for multiplicating and dividing multi-digit numbers were developed.

Irrational numbers were already considered by Pisa and were later discussed more thoroughly by S. dal Ferro and N. Tartaglia in the 16th century.

First attempts to deal with complex numbers were undertaken by [R. Bombelli] in the 16th century but a precise treatment was not given until the works of A. de Moivre and L. Euler in the 18th century.

While the idea of logarithms was present as early as the 3rd century BCE in the works of Archimedes, their treatment was only properly developed in the 16th and 17th centuries by Stifel, J. Napier, and J. Burgi.

Calculating machines were first developed in the 17th century by W. Schickard and B. Pascal. They came to be widely used starting in the 19th century and became very common in the middle of the 20th century following the development of electronic computers.

In other field

Psychology

numeracy

Philosophy

A further topic in the philosophy of arithmetic concerns its ontological foundations. It concerns questions like what type of entities the numbers studied by arithmetic are. According to Platonism, they are abstract objects existing independent of space and time and without any causal powers. Different views hold that they are mental objects existing in the minds of thinkers or that they are features of physical objects.

Many problems in the philosophy of arithmetic lie at the core of discussion in the philosophy of mathematics in general.

Many of the difficulties in the philosophy of arithmetic arise from the observation that several of its aspects are in tension with each other. Its claims are objective, i.e., independent of one's cultural background or personal preferences. At the same time, it is know through thinking alone and does not require perceptual experience to verify its claims. It also has a specific subject matter: it studies numbers.

The objectivity of arithmetic could be explained either in terms of the objectivity of pure reason or in terms of its subject matter. However, it is not clear how both of these explanations can be true since it is not clear how pure thinking can arrive at truths about a subject matter that exists independent of thought.

One solution to this problem was proposed by Frege, who held that the objectivity of arithmetic arises from the objectivity of pure thinking by being grounded in the laws of logic. In this regard, numbers are not independent entities but only secondary phenomena that emerge when when trying to express arithmetic truths with the use of singular terms.

Another solution is presented by the thesis of fictionalism. It states that numbers are fictional entities. According to this view, arithmetic claims are not literally true but are internally consistent inventions that can be used to make predictions and solve problems.

Another issue in the philosophy of arithmetic concerns the question of what type of entities numbers are.

Numbers are not ordinary objects like cups since it is possible to destroy ordinary objects but it is not possible to destroy numbers.

Numbers are used to count and quantify things.

One question in the philosophy of mathematics is to explain why numbers and arithmetic operations can be used to describe the empirical world. For example, if a person has 4 apples and receives 3 additional apples, then arithmetic can be used to infer that they now have 7 apples. A closely related problem is how this can be known a priori, i.e. without relying on empirical observations by counting the apples.

According to an influential hypothesis by Gottlob Frege, numbers are closely related to the sizes of collections. According to one of his basic laws, the number of entities belonging to one collection is identical to the number of entities belonging to another collection if and only if there is a one-to-one correspondence between the entities in the collections.

The application problem concerns the question of why numbers can be used to count things and why arithmetic operations can successfully predict the quantities of empirical entities.

According to one view, number word are not singular terms referring to entities but are instead determiners that

One contrast in the philosophy of arithmetic is between the adjectival strategy and the substantival strategy. The substantival strategy understand number words as singular terms while the adjective strategy sees numbers as properties or descriptors of sets or groups of objects.

According to logicism, the laws of arithmetic can be reduced to the laws of logic.

Computer and cryptography

rel to cryptography

RSA

Other areas of mathematics and the sciences

Arithmetic belongs to the foundations of many other branches of mathematics.

Arithmetic plays a central role in algebra, which is concerned with

Everyday life

Arithmetic has many practical applications in everyday life. For example, it is needed to calculate what change one receives after a purchase.

Assorted

quote: " study of the distribution of prime numbers; the two main theorems are the prime number theorem: “the number of prime numbers smaller than x is asymptotically equivalent to x/ log x” and the theorem on arithmetic progressions: “there are infinitely many prime numbers congruent to m modulo n when m and n are relatively prime”."

Complex numbers expand the set of real numbers with imaginary numbers. Imaginary numbers are defined by the square root of negative numbers and denoted as 'i'. The symbol for the set of real numbers is C.

A special type of numbers are prime numbers. A prime number is a natural number that has only two distinct positive divisors: 1 and itself. This means that it cannot formed by multiplying any other natural numbers. Examples of prime numbers include 2, 3, 5, 7, 11, and 13. 6, for instance, is not a prime number since it can be divided by 2 and 3.

References

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